FIXED POINT THEOREM IN STRUCTURE FUZZY METRIC SPACE

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1 International Journal of Pure and Applied Mathematics Volume 99 No , ISSN: (printed version); ISSN: (on-line version) url: doi: PAijpam.eu FIXED POINT THEOREM IN STRUCTURE FUZZY METRIC SPACE Santanu Acharjee Mathematical Sciences Division Institute of Advanced Study in Science and Technology Paschim Boragaon, Guwahati, , Assam, INDIA Abstract: In this short paper we prove a fixed point theorem on structure fuzzy metric space which doesn t have unique fixed point. AMS Subject Classification: 54H25, 47H10 Key Words: structure fuzzy metric space 1. Introduction After Zadeh [5] introduced fuzzy sets in 1965, many researchers from various areas have developed the theory of fuzzy sets and its applications. Deng [8],Erceg [9], George and Veeramani [7] etc gave initial foundations of different forms of fuzzy metric spaces. Grebiec[3] extended Banach s[11] and Edelstein s[13] fixed point theorem in fuzzy metric space. Kramosil and Michalek [12] investigated common fixed point theorems for compatible maps. In this paper we will study a fixed point theorem from view point of a new class of fuzzy metric defined on a set. Fixed point theorems in any areas are most useful. Mathematical economists, physicists, computer scientists etc are using fixed poind theorems in their respective research. At present various types of fuzzy fixed point theorems are also playing crucial role in mathematical economics, social choices, auction theory. Received: December 4, 2014 c 2015 Academic Publications, Ltd. url:

2 368 S. Acharjee 2. Preinary Definitions In this section we discuss some existing definitions. Definition 2.1. (see [6]) A binary operation : [0,1] [0,1] [0,1] is a continuous t-norm if * satisfies the following conditions: (a) * is commutative and associative; (b) * is continuous; (c) a 1 = a a [0,1]; (d) a b c d whenever a c and b d and a,b,c,d [0,1]. Definition 2.2. (see [12]) Let X be a non-empty set, * be a continuous t-norm and M : X 2 [0, ) [0,1] be a fuzzy set. Consider the following conditions for all x,y,z X and t,s [0, ): (M1) M(x,y,0) = 0 (M2) M(x,x,t) = 1 (M3) M(x,y,t) = 1 x = y (M4) M(x,y,t) = M(y,x,t) (M5) M(x,y,t+s) M(x,z,t) M(z,y,s) (M6) M(x,y,.) : [0, ) [0,1] is left continuous Then (X,M, ) is said to be a fuzzy metric space. 3. Main Result In this section we will prove a fixed point theorem which doesn t have unique fixed point by removing certain conditions from definitions of fuzzy metric space. Definition 3.1. (X,M, ) is said to be a structure fuzzy metric space (SFMS) if it satisfies conditions (M1),(M3),(M4),(M5) and (M6) of Definition 2.2. Definition 3.2. A sequence < x n > in a SFMS is said to be structure convergent if x X such that n M(x n,x,t) = 1 t > 0. Then x is said to be structure it of < x n > and denoted by n x n=x.

3 FIXED POINT THEOREM IN Definition 3.3. A sequence < x n > in a SFMS (X,M, ) is said to be structure cauchy sequence if for each t > 0 and r N such that n M(x n+r,x n,t) = 1. (X,M, ) issaidtobestructurecompleteifeverystructurecauchy sequence in it is structure convergent. Definition 3.4. Let (X,M, ) be a SFMS and f and h are self maps on X. Thenf andhare said tobenormalizedat x if andonly ifm(fhx,hfx,t) = 1 t [0, ). f and h are said to be normalized on X if f and h are normalized at all point x of X. Definition 3.5. f and h are said to be common domain normalized (CDN) if they are normalized at the coincidence point of f and h. Theorem 3.1. Let (X,M, ) be a SFMS, f,h : X X be two mappings and D be the set of all coincidence points of f and h with the following conditions: 0 (a) f(x) h(x) (b) Either f(x) or h(x) is structure complete (c) M(fx,fy,t) g(m(hx,hy,t)) x,y X, t (0, ) and M(hx,hy,t) > (d) g : (0,1] (0,1] is monotonic increasing and n gn (β) = 1,g(β) β β (0,1] (e) M(hx,fx,t) > 0 t > 0 and for some fixed x X Then f and h have a coincidence point; moreover if f and h are CDN and M(h 2 a,ha,t) > 0 for some a D and t (0, ) then f and h have a fixed point. Proof. Let x o be fixed and x 0 X and M(hx 0,fx o,t) > 0 t (0, ). As f(x) g(x) then x 1 X such that y 1 = fx o = hx 1. By mathematical induction we have, y n+1 = fx n = hx n+1 n N and y o = hx o. Now M(y n+1,y n+2,t) g n (M(hx o,fx o,t)) n N,t (0, ). As n, so by condition (d) we have M(y n+1,y n+2,t) 1. Thus M(y n,y n+2,t) M(y n,y n+1, t 2 ) M(y n+1,y n+2, t 2 ),

4 370 S. Acharjee M(y n,y n+3,t) M(y n,y n+1, t 3 ) M(y n+1,y n+2, t 3 ) M(y n+2,y n+3, t 3 ). Proceeding by using mathematical induction we have the following step M(y n,y n+r,t) M(y n,y n+1, t r ) M(y n+1,y n+2, t r ) M(y n+2,y n+3, t r )... M(y n+r 1,y n+r, t r ), where r N. If n then M(y n,y n+r,t) 1, i.e. n M(y n,y n+r,t) = 1. Thus implies < y n > is a structure Cauchy sequence. Let us assume that h(x) is structure complete, then u h(x) such that n y n+1= n hx n+1=u= n fx n. Let hv = u for some v X. If n then by definition of SFMS,we have M(hv,fx n,t) 1 M(hv,hx n+1,t) 1. By condition (c) and (d), we have M(fv,fx n+1,t) g(m(hv,hx n+1,t)) M(hv,hx n+1,t). If n then we have fv = hv i.e. f and h have a coincidence point. Let us consider that f and h are CDN and M(h 2 a,ha,t) > 0 for some a D and t (0, ). Let fa = ha = ω. So, by condition (c), M(fha,fa,t) g(m(h 2 a,ha,t)) > 0. It implies M(hfa,ha,t) > 0, after proceeding with mathematical induction we have M(f 2 a,fa,t) g n (M(hfa,ha,t)). As n then by condition (d), M(f 2 a,fa,t) 1. It implies that f 2 a = fa or fω = ω. Now easy to verify that hfa = fa or hω = ω. Hence ω is the fixed point of f and h. In similar way we can prove for structure completeness of f(x). Remark 3.1. It can be easily checked that the case of uniqueness of fixed point is not possible in the above fixed point theorem. Acknowledgments Author is very much thankful to his mother Mrs. M. Acharjee for her constant support and inspiration in author s life. Author is also very thankful to reviewers of this research paper for their suggestions. References [1] J.X. Feng, On fixed point theorem in fuzzy metric space, Fuzzy sets and Systems, 46(1992), , doi: / (92)

5 FIXED POINT THEOREM IN [2] K. Kuratowski, Topology-I, Polish Scientific Publishers, Warsaw (1933). [3] M. Grabeic, Fixed points in fuzzy metric spaces, Fuzzy sets and Systems, 27 (1988), , doi: / (88) [4] R. Vasuki, Common fixed point theorem in fuzzy metric space, Fuzzy sets and Systems, 97 (1998), , doi: /S (96) [5] L.A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965), , doi: /S (65)90241-X. [6] B. Schweizer and A. Sklar, Statistical metric space, Pacific Jour. Math, 10 (1960), ( [7] A. George and P. Veeramani, On some results in fuzzy metric space, Fuzzy sets and Systems, 64 (1994), , doi: / (94) [8] Z.K. Deng, Fuzzy pseudo-metric space, J. Math. Anal. Appl, 86, (1982), , doi: / X(82) [9] M.A. Erceg, Metric spaces in fuzzy set theory, J. Math. Anal. Appl, 69, (1979), (doi: / x(79) ). [10] Y.J. Cho, Fixed point in fuzzy metric space, J. Fuzzy math, 5 (1997), [11] S. Banach, Theories les operation linearies, Manograie Mathematyezne, (1932). [12] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric space, Kybernatika,11 (1975), [13] M. Edelstein, On fixed and periodic points under contraction mapping, J. London Math. Soc, 37, (1962),74-79 (doi: /jlms/s ). [14] J.F. Nash, Equilibrium points in n-person games, PNAS, 36, (1950), 48-49, doi: /pnas [15] S. Kakutani, A generalization of Brouwer s fixed point theorem, Duke Math. Jour, 8, (1941), (doi: /s ). [16] S. Acharjee, Fixed point theorem in fuzzy metric space, Boletim da Sociedade Paranaense de Matematica, accepted (December, 2014).

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